Problem:
What is the least number that must be added to 4131 so that the sum is completely divisible by 19?

Explanation:
To solve this problem, we need to find the smallest number that, when added to 4131, makes the sum divisible by 19.

Method:
One way to approach this problem is to use the concept of congruence. Two numbers are said to be congruent modulo n if their difference is divisible by n.

Congruence:
In this case, we want to find a number x such that (4131 + x) is congruent to 0 modulo 19.

Modulo calculation:
To find the congruence, we can use the modulo operation. Taking the modulo of a number gives us the remainder when the number is divided by the chosen modulus.

Calculating modulo:
In this case, we want to find the remainder when (4131 + x) is divided by 19. So, we can write the congruence as:
(4131 + x) ≡ 0 (mod 19)

Calculating the remainder:
To find the remainder, we can divide 4131 by 19 and take the remainder:
4131 ÷ 19 = 217 remainder 8

Equation:
Now, let's substitute the remainder into the congruence equation:
(8 + x) ≡ 0 (mod 19)

Solving the congruence equation:
To solve this congruence equation, we need to find the value of x that satisfies the equation. We can do this by finding the smallest positive value of x that satisfies the equation.

Calculating x:
To find the value of x, we can start substituting values for x and checking if the equation holds true. We can start with x = 1 and continue until we find a value that satisfies the congruence.

Checking with x = 1:
When x = 1, the congruence equation becomes:
(8 + 1) ≡ 0 (mod 19)
9 ≡ 0 (mod 19)

Checking with x = 2:
When x = 2, the congruence equation becomes:
(8 + 2) ≡ 0 (mod 19)
10 ≡ 0 (mod 19)

Checking with x = 3:
When x = 3, the congruence equation becomes:
(8 + 3) ≡ 0 (mod 19)
11 ≡ 0 (mod 19)

Finding the smallest value:
We can continue checking values until we find the smallest value that satisfies the congruence equation.

Answer:
From our calculations, we can see that the smallest value of x that satisfies the congruence equation is x = 11. Therefore, the least number that must be added to 4131 to make the sum completely divisible by 19 is 11.